Symmetry, Integrability and Geometry: Methods and Applications
SIGMA 6 (2010), 070, 17 pages
C-Integrability Test for Discrete Equations
via Multiple Scale Expansions
Christian SCIMITERNA and Decio LEVI
Dipartimento di Ingegneria Elettronica, Università degli Studi Roma Tre and Sezione INFN,
Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy
E-mail: scimiterna@fis.uniroma3.it, levi@roma3.infn.it
Received May 29, 2010, in final form August 20, 2010; Published online August 31, 2010
doi:10.3842/SIGMA.2010.070
Abstract. In this paper we are extending the well known integrability theorems obtained by
multiple scale techniques to the case of linearizable difference equations. As an example we
apply the theory to the case of a differential-difference dispersive equation of the Burgers
hierarchy which via a discrete Hopf–Cole transformation reduces to a linear differential
difference equation. In this case the equation satisfies the A1 , A2 and A3 linearizability
conditions. We then consider its discretization. To get a dispersive equation we substitute
the time derivative by its symmetric discretization. When we apply to this nonlinear partial
difference equation the multiple scale expansion we find out that the lowest order nonsecularity condition is given by a non-integrable nonlinear Schrödinger equation. Thus
showing that this discretized Burgers equation is neither linearizable not integrable.
Key words: linearizable discrete equations; linearizability theorem; multiple scale expansion;
obstructions to linearizability; discrete Burgers
2010 Mathematics Subject Classification: 34K99; 34E13; 37K10; 37J30
1
Introduction
Calogero in 1991 [3] introduced the notion of S and C integrable equations to denote those
nonlinear partial differential equations which are solvable through an inverse Spectral transform
or linearizable through a Change of variables. Using the multiple scale reductive technique he
was able to show that the Nonlinear Schrödinger Equation (NLSE)
i∂t f + ∂xx f = ρ2 |f |2 f,
f = f (x, t),
(1.1)
appears as a universal equation governing the evolution of slowly varying packets of quasimonochromatic waves in weakly nonlinear media featuring dispersion. Calogero and Eckhaus
then showed that a necessary condition for the S-integrability of a dispersive nonlinear partial
differential equation is that its multiple scale expansion around a slowly varying packet of
quasi-monochromatic wave should provide at the lowest order in the perturbation parameter an
integrable NLSE. Then, by a paradox, they showed that a C-integrable equation must reduce
to a linear equation or another C-integrable equation as the Eckhaus equation [4, 22].
In the case of discrete equations it has been shown [26, 1, 16, 18, 17, 19, 10, 11, 12] that
a similar situation is also true. One presents the equivalent of the Calogero–Eckhaus theorem
stating that a necessary condition for a nonlinear dispersive partial difference equation to be
S-integrable is that the lowest order multiple scale expansion on C (∞) functions give rise to
integrable NLSE. The nonlinear dispersive partial difference equation will be C-integrable if
its multiple scale expansion on C (∞) functions will give rise at the lowest order to a linear or
a C-integrable differential equation.
By going to a higher order in the expansion, the multiple scale techniques give more stringent
conditions which have been used to find new S-integrable Partial Differential Equations (PDE’s)
2
C. Scimiterna and D. Levi
and to prove the integrability of new nonlinear equations [8, 9, 15]. Probably the most important
example of such nonlinear PDE is the Degasperis–Procesi equation [7]. Up to our knowledge
higher order expansions for nonlinear linearizable equations have not been considered in details.
The purpose of this paper is to show that the integrability theorem, stated in [11], can
be extended to the case of linearizable difference equations, providing a way to discriminate
between S-integrable, C-integrable and non–integrable lattice equations. The continuous (and
thus discrete) higher order C-integrability conditions are, up to our knowledge, presented here for
the first time. We apply here the resulting linearizability conditions to a differential-difference
dispersive nonlinear equation of the discrete Burgers hierarchy [21] and its difference-difference
analogue.
In Section 2 we present the differential-difference linearizable nonlinear dispersive Burgers
and its partial difference analogue and discuss the tools necessary to carry out the multiple scale
C-integrability test. In Section 3 we apply them to the two equations previously introduced,
leaving to the Appendix all details of the calculation, and present in Section 4 some conclusive
remarks.
2
Multiple scale perturbation reduction of Burgers equations
The Burgers equation, the simplest nonlinear equation for the study of gas dynamics with heat
conduction and viscous effect, was introduced by Burgers in 1948 [2, 27]. Explicit solutions of
the Cauchy problem on the infinite line for the Burgers equation may be obtained by the Hopf–
Cole transform, introduced independently by Hopf and Cole in 1950 [5, 14]. This transformation
linearize the equation and the solution of the linearized equation provide solutions of the Burgers
equation.
Bruschi, Levi and Ragnisco [21] extended the Hopf–Cole transformation to construct hierarchies of linearizable nonlinear matrix PDE’s, nonlinear differential-difference equations and
difference-difference equations.
The simplest differential–difference nonlinear dispersive equation of the Burgers hierarchy is
given by
1
un−1 (t) − un (t)
∂t un (t) =
[1 + hun (t)] [un+1 (t) − un (t)] −
,
(2.1)
2h
1 + hun−1 (t)
where the function un (t) and the lattice parameter h are all supposed to be real. Equation (2.1)
has a nonlinear dispersion relation ω = − sin(κh)
.
h
When h → 0 and n → ∞ in such a way that x = nh is finite, the Burgers equation (2.1)
reduces to the one dimensional wave equation ∂t u − ∂x u = O(h3 ).
Through the discrete Cole–Hopf transformation
un (t) =
φn+1 (t) − φn (t)
hφn (t)
(2.2)
equation (2.1) linearizes to the discrete linear wave equation
∂t φn (t) =
φn+1 (t) − φn−1 (t)
.
2h
(2.3)
The transformation (2.2) can be inverted and gives
φn = φn0
j=n−1
Y
j=n0
(1 + huj ) ,
n ≥ n0 + 1,
(2.4a)
C-Integrability Test for Discrete Equations via Multiple Scale Expansions
φn =
φn0
j=n
Q0 −1
,
n ≤ n0 − 1,
3
(2.4b)
(1 + huj )
j=n
where φn0 = φn0 (t) is the function φn calculated at a given initial point n = n0 . When un (t)
satisfies equation (2.1), the function φn (t) will satisfy the discrete wave equation (2.3) if φn0
satisfies the ordinary differential equation
1
1
= 0,
φ̇n −
1 + hun −
φn 2h
(1 + hun−1 )
n=n0
whose solution is given by
Z t
1
1
0
φn0 (t) = φn0 (t0 ) exp
1 + hun −
dt ,
2h t0
(1 + hun−1 ) n=n0
and t0 is an initial time. If lim un (t) = u−∞ is finite equations (2.4) reduce to
n→−∞
n
φn = α (t) (1 + hu−∞ )
γ=n−1
Y γ=−∞
1 + huγ
1 + hu−∞
,
where α (t) is a t-dependent function,
1
1
α(t) = α0 exp
1+
u−∞ t .
2
1 + hu−∞
A C-integrable discretization of equation (2.1) is given by the partial difference equation
un,m+1 − un,m
un−1,m − un,m+1
1
=
(1 + hun,m )(un+1,m − un,m+1 ) −
,
(2.5)
σ
2h
1 + hun−1,m
where σ is the constant lattice parameter in the time variable. Equation (2.5) is dissipative
as it has a complex dispersion relation ω = σi ln 1 + i σh sin (κh) . As we are not able to construct a dispersive counterpart of equation (2.1) we consider a straightforward discretization of
equation (2.1)
un,m+1 − un,m−1
(un−1,m − un,m )
1
=
(1 + hun,m ) (un+1,m − un,m ) −
,
(2.6)
2σ
2h
1 + hun−1,m
whose nonlinear dispersion relation is
sin (ωσ) = −
sin (κh) σ
.
h
In the remaining part of this section we will present the tools necessary to carry out the
multiple scale expansion of equations (2.1), (2.6) and construct the conditions which they must
satisfy to be C-integrable equation of j order, with j = 1, 2, 3, i.e. such that asymptotically they
reduce to a linear equation up to terms respectively of the third, fourth and fifth order in the
perturbation paramether. These conditions up to our knowledge have been presented for the
first time by Dr. Scimiterna in his PhD Thesis [25] and are published here for the first time.
2.1
Expansion of real dispersive partial dif ference equations
For completeness we briefly illustrate here all the ingredients of the reductive perturbative
technique necessary to treat difference equations, as presented in [10, 11, 25].
4
C. Scimiterna and D. Levi
2.1.1
From shifts to derivatives
Let us consider a function un : Z → R depending on an index n ∈ Z and let us suppose that:
.
• The dependence of un on n is realized through the slow variable n1 = n ∈ R, ∈ R,
.
0 < 1, that is to say un = u(n1 ).
• The function u (n1 ) ∈ C (∞) (D), where D ∈ R is a region containing the point n1 .
.
Under these hypotheses one can write the action of the shift operator Tn such that Tn un =
un+1 = u(n1 + ) as the following (formal) series
Tn u(n1 ) = u(n1 ) + u,n1 (n1 ) +
=
+∞ k
X
k=0
k!
2
k
u,2 n1 (n1 ) + · · · + u,k n1 (n1 ) + · · ·
2
k!
u,k n1 (n1 ),
(2.7)
.
.
where u,k n1 (n1 ) = dk u(n1 )/dnk1 = dkn1 u(n1 ), being dn1 the total derivative operator. The last
expression suggests the following formal expansion for the differential operator Tn :
Tn =
+∞ k
X
k=0
k!
.
dkn1 = edn1
valid only when the series in equation (2.7) is converging. So we must require that the radius
of convergence of the series starting at n1 is wide enough to include as an inner point at least
the point n1 + .
Let us introduce more complicated dependencies of un on n. For example one can assume
.
a simultaneous dependence on the fast variable n and on the slow variable n1 , i.e. un = u(n, n1 ).
.
The action of the total shift operator Tn will now be given by Tn un = un+1 = u(n + 1, n1 + )
. (1) ()
(1)
()
so that we can write Tn = Tn Tn1 , where the partial shift operators Tn and Tn1 are defined
respectively by
Tn(1) u(n, n1 ) = u(n + 1, n1 ) =
∞
X
1 k
∂ u(n, n1 ) = e∂n u(n, n1 ),
k! n
k=0
and
Tn()
u(n, n1 ) = u(n, n1 + ) =
1
∞ k
X
k=0
k!
∂nk1 u(n, n1 ) = e∂n1 u(n, n1 ).
(2.8)
The dependence of un on n can be easily extended to the case of one fast variable n and K slow
.
variables nj = j n, j ∈ R, 1 ≤ j ≤ K each of them being defined by its own parameter j . The
(1)
()
action of the total shift operator Tn will now be given in terms of the partial shifts Tn , Tnj ,
K
. (1) Q
( )
as Tn = Tn
Tnj j .
j=1
Let us now consider a nonlinear partial difference equation
j=(−K(−) ,K(+) )
F {un+k,m+j }k= −N (−) ,N (+) = 0,
N (±) , K(±) ≥ 0,
(
)
(2.9)
for a function un,m : Z2 → R which now depends on two indexes n and m ∈ Z which we will
term respectively as discrete space and time indices. Equation (2.9) contains m and n-shifts,
respectively in the intervals (m − K(−) , m + K(+) ) and (n − N (−) , n + N (+) ). Under some obvious
C-Integrability Test for Discrete Equations via Multiple Scale Expansions
(j)
(j)
5
(j)
Table 1. The operators An , Bm and Cn,m appearing in equations (2.10).
j=0
(j)
An
(j)
Bm
j=1
j=2
j=3
j=4
1
N1 ∂n1
N12 2
∂n1
2
N13 3
∂n1
6
N14 4
∂
24 n1
1
M1 ∂m1
M13 3
∂m1 +
6
M 2M
M14 4
2
∂ + 12 2 ∂m
∂ +
1 m2
24 m1
M22 2
+ 2 ∂m2 + M1 M3 ∂m1 ∂m3 + M4 ∂m4
M12 2
∂m1
2
+ M2 ∂m2
+M1 M2 ∂m1 ∂m2 +
+M3 ∂m3
(j)
Cn,m
(1)
(1)
An + Bm
1
(2)
(2)
(3)
(3)
An + Bm +
An + Bm +
+N1 M1 ∂n1 ∂m1
+N1 M2 ∂n1 ∂m2 +
M1 N12 2
∂n1 ∂m1 +
2
N1 M12
2
+ 2 ∂n1 ∂m
1
+
(4)
(4)
An + Bm +
M13 N1 3
N13 M1 3
+ 6 ∂m1 ∂n1 + 6 ∂n1 ∂m1 +
N 2M
+N1 M1 M2 ∂n1 ∂m1 ∂m2 + 12 2 ∂n2 1 ∂m2 +
N 2M 2
2
+ N1 M3 ∂n1 ∂m3
+ 14 1 ∂n2 1 ∂m
1
hypothesis on the C (∞) property of the function un,m and on the radius of convergence of its
Taylor expansion for all shifts in the indices n and m involved in the difference equation (2.9),
we can write a series representation of un+k,m+j around un,m . We choose the slow variables as
.
.
nk = nk n, mj = mj m with
.
nk = Nk k ,
.
mj = Mj j ,
1 ≤ k ≤ Kn ,
1 ≤ j ≤ Km ,
where the various constants Nk , Mj and are all real numbers. In this presentation we assume
Kn = 1 and Km = K (eventually K = +∞) so that
(n )
Tn(1) Tn1 1
Tn =
=
Tn(1)
+∞
X
j A(j)
n ,
(2.10a)
j=0
Tm =
Tm(1)
K
Y
(m )
Tmj j
=
Tm(1)
j=1
Tn Tm =
+∞
X
(j)
j Bm
,
(2.10b)
j=0
(n )
Tn(1) Tm(1) Tn1 1
K
Y
(m )
Tmj j
=
j=1
(j)
(j)
Tn(1) Tm(1)
+∞
X
(j)
j Cn,m
,
(2.10c)
j=0
(j)
where the operators An , Bm , and Cn,m are given in Table 1. Inserting the explicit expressions (2.10) of the shift operators into equation (2.9), this turns into a PDE of infinite order. So
we will assume for the function un,m = u(n, m, n1 , {mj }K
j=1 , ) a double expansion in harmonics
and in the perturbative parameter un,m =
γ
+∞ X
X
iθ(κhn−ω(κ)σm)
γ u(θ)
,
γ (n1 , mj , j ≥ 1) e
(2.11)
γ=1 θ=−γ
(−θ)
(θ)
with uγ (n1 , mj , j ≥ 1) = ūγ (n1 , mj , j ≥ 1), where by a bar we denote the complex conjugate,
in order to ensure the reality of un,m . The index γ is chosen ≥ 1 so that the nonlinear terms
of equation (2.9) enter as a perturbation in the multiple scale expansion. For simplicity we will
(θ)
set N1 = Mj = 1, j ≥ 1. Moreover we will assume that the functions uγ satisfy the asymptotic
(θ)
conditions lim uγ = 0, ∀ γ and θ to provide a meaningful expansion.
n1 →±∞
2.1.2
From derivatives to shifts
The multiple scale approach discussed above reduces a given partial difference equation into
(θ)
a partial differential equation for the amplitudes uγ contained in the definition (2.11).
6
C. Scimiterna and D. Levi
We can rewrite the so obtained partial differential equation as a partial difference equation inverting the expansion of the partial shift operator in term of partial derivatives (2.8).
From (2.8) we have
+∞
∂n1 =
X (−)k−1 (+) k
1
1
(+) .
∆n1 ,
ln Tn()
=
ln
1
+
∆
=
n1
1
k
(2.12)
k=1
()
(+) .
where ∆n1 = (Tn1 − 1)/ is the first forward difference operator with respect to the slow()
variable n1 . This is just one of the possible inversion formulae for the operator Tn1 . For example
an expression similar to equation (2.12) can be written for the first backward difference operator
() −1 () () −1 (s) .
(−) .
/2
/. For the first symmetric difference operator ∆n1 = Tn1 − Tn1
∆n1 = 1− Tn1
we get
+∞
. X Pk−1 (0)k (s) k
=
∂n1 = sinh−1 ∆(s)
∆n1 ,
n1
k
k=1
where Pk (0) is the k-th Legendre polynomial evaluated at x = 0.
Only when we impose that the function un is a slow-varying function of order ` in the
variable n1 , i.e. that ∆`+1
n1 un = 0, we can see that the ∂n1 operator, which is given by formal
series containing in general infinite powers of the ∆n1 , reduces to polynomial of order at most `.
In [17], choosing ` = 2 for the indexes n1 and m1 and ` = 1 for m2 , it was shown that the
integrable lattice potential KdV equation [20] reduces to a completely discrete and local nonlinear
Schrödinger equation which has been proved to be not integrable by singularity confinement and
algebraic entropy [13, 23]. Consequently, if one passes from derivatives to shifts, one ends up in
general with a nonlocal partial difference equation in the slow variables nκ and mδ .
2.2
The orders beyond the Schrödinger equation
and the C-integrability conditions
The multiple scale expansion of an equation of the Burgers hierarchy on functions of infinite
order will thus give rise to PDE’s. So a multiple scale integrability test will require that a dispersive equation like equation (2.1) is C-integrable if its multiple scale expansion will go into
the hierarchy of the Schrödinger equation
i∂t ψ + ∂x2 ψ = 0.
To be able to verify the C-integrability we need to consider in principle all the orders beyond
the Schrödinger equation. This in general will not be possible but already a few orders beyond
the Schrödinger equation might be sufficient to verify if the equation is linearizable or not. In
the case of S-integrable nonlinear PDE’s the first attempt to go beyond the NLSE order has
been presented by Degasperis, Manakov and Santini in [8]. These authors, starting from an
S-integrable model, through a combination of an asymptotic functional analysis and spectral
methods, succeeded in removing all the secular terms from the reduced equations order by order.
Their results could be summarized as follows:
(θ)
1. The number of slow-time variables required for the amplitudes uj appearing in (2.11)
coincides with the number of nonvanishing coefficients of the Taylor expansion of the
j ω(k)
dispersion relation, ωj (κ) = j!1 d dk
j .
(1)
2. The amplitude u1 evolves at the slow-times ms , s ≥ 2 according to the s-th equation of
the NLS hierarchy.
C-Integrability Test for Discrete Equations via Multiple Scale Expansions
7
(1)
3. The amplitudes of the higher perturbations of the first harmonic uj , j ≥ 2 evolve, taking
into account some asymptotic boundary conditions, at the slow-times ms , s ≥ 2 according
to certain linear, nonhomogeneous equations.
Then they concluded that the cancellation at each stage of the perturbation process of all
the secular terms is a sufficient condition to uniquely fix the evolution equations followed by
(1)
every uj , j ≥ 1 for each slow-time ms . Point 2 implies that a hierarchy of integrable equations
provide for a function u always compatible evolutions, i.e. the equations in its hierarchy are
generalized symmetries of each other. In this way this procedure provides necessary and sufficient
conditions to get secularity-free reduced equations [8].
We apply the present procedure to the case of C-integrable partial difference equations.
Following Degasperis and Procesi [9] we state the following theorem:
Theorem 1. If equation (2.9) is C-integrable then, after a multiple scale expansion, the func(1)
tions uj , j ≥ 1 satisfy the equations
(1)
(1) .
(1)
∂ms u1 − (−i)s−1 Bs ∂ns 1 u1 = Ms u1 = 0,
(1)
Ms u j
(2.13a)
= fs (j),
(2.13b)
(j)
∀ j, s ≥ 2, where Bs ∂ns 1 u1 is the s-th f low in the linear Schrödinger hierarchy and Bs are real
(κ)
(1)
constants. All the other uj , κ ≥ 2 are expressed as differential monomials of ur , r ≤ j − 1.
In equation (2.13b) fs (j) is a nonhomogeneous nonlinear forcing written in term of differential
(1)
monomials of ur , r ≤ j. From Theorem 1 it follows that a nonlinear partial difference equation
is said to be C-integrable if its asymptotic multiple scale expansion is given by a uniform
asymptotic series whose leading harmonic u(1) possesses an infinity of generalized symmetries
evolving at different times and given by commuting linear equations. Equations (2.13) are
a necessary condition for C-integrability.
It is worthwhile to stress here the non completely obvious fact that, in contrast to the first
order wave equation, ∂t u = ∂x u, all the symmetries of the Schrödinger equation commuting
with it are given by the equations (2.13a) and only by them. This implies that all the equations
appearing in the multiple scale expansion for a C-integrable equation are uniquely defined.
It is obvious that the operators Ms defined in equation (2.13a) commute among themselves.
However the compatibility of equations (2.13b) is not always guaranteed but is subject to some
compatibility conditions among their r.h.s. terms fs (j). Once we fix the index j ≥ 2 in the set
of equations (2.13b), this commutativity condition implies the compatibility conditions
Ms fs0 (j) = Ms0 fs (j) ,
∀ s, s0 ≥ 2,
(2.14)
(1)
where, as fs (j) and fs0 (j) are functions of the different perturbations ur of the fundamental
harmonic up to degree j −1, the time derivatives ∂ms , ∂ms0 appearing respectively in Ms and Ms0
have to be eliminated using the evolution equations (2.13) up to the index j − 1. These last
commutativity conditions turn out to be a linearizability test.
Following [8] we conjecture that the relations (2.13) are a sufficient condition for the Cintegrability or that the C-integrability is a necessary condition to have a multiple scale expansion where equations (2.13) are satisfied. To characterize the functions fs (j) we introduce the
following definitions:
(1) (1)
Definition 1. A differential monomial M uj , j ≥ 1 in the functions uj , its complex conjugate and its n1 -derivatives is a monomial of “gauge” 1 if it possesses the transformation property
(1) (1) (1) .
(1)
M ũj = eiθ M uj ,
when ũj = eiθ uj .
8
C. Scimiterna and D. Levi
Definition 2. A finite dimensional vector space Pν , ν ≥ 2 is the set of all differential polyno(1)
mials of gauge 1 in the functions uj , j ≥ 1, their complex conjugates and their n1 -derivatives
such that their total order in is ν, i.e.
(1) order ∂nµ1 uj
(1) = order ∂nµ1 ūj
= µ + j = ν,
µ ≥ 0.
Definition 3. Pν (µ), µ ≥ 1 and ν ≥ 2 is the subspace of Pν whose elements are differential
(1)
polynomials of gauge 1 in the functions uj , their complex conjugates and their n1 -derivatives
such that their total order is ν and 1 ≤ j ≤ µ.
From Definition 3 it follows that Pν = Pν (ν−2). Moreover in general fs (j) ∈ Pj+s (j−1)
where j, s ≥ 2. The basis monomials of the spaces Pν (µ) in which we can express the functions fs (j) can be found, for example, in [25].
Proposition 1. If for each fixed j ≥ 2 the equation (2.14) with s = 2 and s0 = 3, namely
M2 f3 (j) = M3 f2 (j) ,
(2.15)
is satisfied, then there exist unique differential polynomials fs (j) ∀ s ≥ 4 such that the flows
(1)
Ms uj = fs (j) commute for any s ≥ 2 [24, 6].
Hence among the relations (2.14) only those with s = 2 and s0 = 3 have to be tested.
Proposition 2. The homogeneous equation Ms u = 0 has no solution u in the vector space Pµ ,
i.e. Ker (Ms ) ∩ Pµ = ∅.
Consequently the multiple scale expansion (2.13) is secularity-free. This does not mean that,
in solving equation (2.13b), we have to set to zero all the contributions to the solution coming
from the homogeneous equation but only that part of it which is present in the forcing terms.
Finally:
Definition 4. If the relations (2.14) are satisfied up to the index j, j ≥ 2, we say that our
equation is asymptotically C-integrable of degree j or Aj C-integrable.
2.2.1
Integrability conditions for the Schrödinger hierarchy
Here we present the conditions for the asymptotic C-integrability of order k or Ak C-integrability
(1)
conditions with k = 1, 2, 3. To simplify the notation, we will use for uj the concise form u(j).
The A1 C-integrability condition is given by the absence of the coefficient ρ2 of the nonlinear
term in the NLSE (1.1).
The A2 integrability conditions are obtained choosing j = 2 in the compatibility conditions (2.14) with s = 2 and s0 = 3 as in (2.15). In this case we have that f2 (2) ∈ P4 (1)
and f3 (2) ∈ P5 (1) where P4 (1) contains 2 different differential monomials and P5 (1) contains
5 different differential monomials, so that f2 (2) and f3 (2) will be respectively identified by 2
and 5 complex constants
.
f2 (2) = au,n1 (1)|u(1)|2 + bū,n1 (1)u(1)2 ,
(2.16)
.
4
2
2
2
2
f3 (2) = α|u(1)| u(1) + β|u,n1 (1)| u(1) + γu,n1 (1) ū(1) + δ ū,2n1 (1)u(1) + ε|u(1)| u,2n1 (1).
In this way, eliminating from equation (2.15) the derivatives of u(1) with respect to the slowtimes m2 and m3 using the evolutions (2.13a) with s = 2, 3 and equating term by term, we
obtain that the A2 C-integrability conditions gives no constraints on the coefficients a and b
C-Integrability Test for Discrete Equations via Multiple Scale Expansions
9
appearing in f2 (2). The expression of the coefficients α, β, γ, δ, ε appearing in f3 (2) in terms
of a and b are
α = 0,
β=−
3iB3 b
,
B2
γ=−
3iB3 a
,
2B2
δ = 0,
ε = γ.
The A3 C-integrability conditions are derived in a similar way setting j = 3 in equation (2.15).
In this case we have that f2 (3) ∈ P5 (2) and f3 (3) ∈ P6 (2) where P5 (2) contains 12 different
differential monomials and P6 (2) contains 26 different differential monomials, so that f2 (3)
and f3 (3) will be respectively identified by 12 and 26 complex constants
.
f2 (3) = τ1 |u(1)|4 u(1) + τ2 |u,n1 (1)|2 u(1) + τ3 |u(1)|2 u,2n1 (1) + τ4 ū,2n1 (1)u(1)2
+ τ5 u,n1 (1)2 ū(1) + τ6 u,n1 (2)|u(1)|2 + τ7 ū,n1 (2)u(1)2 + τ8 u(2)2 ū(1)
(2.17)
2
+ τ9 |u(2)| u(1) + τ10 u(2)u,n1 (1)ū(1) + τ11 u(2)ū,n1 (1)u(1) + τ12 ū(2)u,n1 (1)u(1),
.
f3 (3) = γ1 |u(1)|4 u,n1 (1) + γ2 |u(1)|2 u(1)2 ū,n1 (1) + γ3 |u(1)|2 u,3n1 (1) + γ4 u(1)2 ū,3n1 (1)
+ γ5 |u,n1 (1)|2 u,n1 (1) + γ6 ū,2n1 (1)u,n1 (1)u(1) + γ7 u,2n1 (1)ū,n1 (1)u(1)
+ γ8 u,2n1 (1)u,n1 (1)ū(1) + γ9 |u(1)|4 u(2) + γ10 |u(1)|2 u(1)2 ū(2) + γ11 ū,n1 (1)u(2)2
+ γ12 u,n1 (1)|u(2)|2 + γ13 |u,n1 (1)|2 u(2) + γ14 |u(2)|2 u(2) + γ15 u,n1 (1)2 ū(2)
+ γ16 |u(1)|2 u,2n1 (2) + γ17 u(1)2 ū,2n1 (2) + γ18 u(2)ū,2n1 (1)u(1)
+ γ19 u(2)u,2n1 (1)ū(1) + γ20 ū(2)u,2n1 (1)u(1) + γ21 u(2)u,n1 (2)ū(1)
+ γ22 ū(2)u,n1 (2)u(1) + γ23 u,n1 (2)u,n1 (1)ū(1) + γ24 u,n1 (2)ū,n1 (1)u(1)
+ γ25 ū,n1 (2)u,n1 (1)u(1) + γ26 ū,n1 (2)u(2)u(1).
Let us eliminate from equation (2.15) with j = 3 the derivatives of u(1) with respect to the slowtimes m2 and m3 using the evolutions (2.13a) with s = 2, 3 and the same derivatives of u(2)
using the evolutions (2.13b) with s = 2, 3. Equating the remaining terms term by term, the A3
C-integrability conditions turn out to be:
i
[b (τ11 − 2τ6 ) + āτ7 ] ,
4B2
aτ8 = bτ8 = 0,
aτ9 = bτ9 = 0,
b̄ − ā τ12 = (b − a) τ10 .
τ1 = −
b̄τ7 =
āτ12
1
(b − a) (τ11 + τ10 − τ6 ) + āτ7 ,
2
= a (τ10 − τ11 ) + bτ6 + āτ7 ,
(2.18)
Sometimes a and b turn out to be both real. In this case the conditions given in equations (2.18)
becomes:
1
1
[b (I11 − 2I6 ) + aI7 ] ,
I1 = −
[b (R11 − 2R6 ) + aR7 ] ,
4B2
4B2
(b − a) (R11 + R10 − R6 − 2R7 ) = 0,
(b − a) (I11 + I10 − I6 − 2I7 ) = 0,
R1 =
(b − a) R8 = 0,
(b − a) I8 = 0,
a (R12 + R11 − R10 − R7 ) = bR6 ,
(b − a) (R12 − R10 ) = 0,
(b − a) R9 = 0,
(b − a) I9 = 0,
a (I12 + I11 − I10 − I7 ) = bI6 ,
(b − a) (I12 − I10 ) = 0,
where τi = Ri + iIi for i = 1, . . . , 12. The expressions of the γj as functions of the τi are:
3B3
3B3
aτ6 − 4iB2 τ1 + b̄τ12 ,
γ2 =
(bτ6 + āτ7 ) ,
2
4B2
4B22
3iB3 τ3
3iB3 τ2
3iB3 τ4
γ3 = −
,
γ4 = 0, γ5 = −
,
γ6 = −
,
2B2
2B2
B2
γ1 =
(2.19)
10
C. Scimiterna and D. Levi
γ7 = γ5 ,
3iB3 τ9
,
2B2
3iB3 τ6
=−
,
2B2
3iB3 τ8
=−
,
B2
3iB3 τ7
,
=−
B2
γ12 = −
γ16
γ21
γ25
3iB3 τ5
,
γ9 = γ10 = γ11 = 0,
B2
3iB3 τ11
3iB3 τ12
γ13 = −
,
γ14 = 0,
γ15 = −
,
2B2
2B2
3iB3 τ10
γ17 = γ18 = 0,
γ19 = −
,
γ20 = γ15 ,
2B2
γ8 = γ3 −
γ22 = γ12 ,
γ23 = γ16 + γ19 ,
γ24 = γ13 ,
γ26 = 0.
The conditions given in equations (2.18), (2.19) appear to be new. Their importance resides in
the fact that a C-integrable equation must satisfy those conditions.
3
Linearizability of the equations of the Burgers hierarchy
Taking into account the results of the previous section we can carry out the multiple scale
expansion of the equations of the Burgers hierarchy. To do so we substitute the definition (2.11)
into equations (2.1), (2.6) and write down the coefficients of the various harmonics θ and of the
various orders j of . When we deal with the differential-difference equation (2.1), we have to
make the substitutions σm → t, σmi → ti . This transformation implies that in this case the
corresponding coefficients ρ2 and B2 will turn out to be σ-independent. In Appendix we present
all relevant equations and here we just present their results.
Proposition 3. The differential-difference equation (2.1) of the Burgers hierarchy satisfies
the A1 (and consequently also the A2 ) and also the A3 C-integrability conditions.
Proposition 4. The partial difference Burgers-like equation (2.6) reduces for j = 3 and α = 1
to a NLSE with a nonlinear complex coefficient ρ2 given by equation (A.9c). Thus the equation
is neither S-integrable nor C-integrable.
4
Conclusions
In the present paper we have presented all the steps necessary to apply the perturbative multiple
scale expansion to dispersive nonlinear differential-difference or partial difference equations
which may be linearizable. These passages involve the representation of the lattice variables in
terms of an infinite set of derivative with respect to the lattice index and the analysis of the
higher order of the perturbation which give rise to a set of compatible higher order linear PDE’s
belonging to the hierarchy of the Schrödinger equation. The compatibility of these equations
give rise to a linearizability test. We applied the so obtained test to the case of a differentialdifference dispersive Burgers equation and its discretization. It turns out that the Burgers
is linearizable (as it should be) but its discretization is neither S-integrable nor C-integrable.
So, effectively this procedure is able to distinguish between linearizable and non-linearizable
equations.
A
Appendix
Let us now start performing a multiple scale analysis of the partial difference equation (2.6). We
present here the equations we get at the various orders of and for the different harmonics θ.
• Order and θ = 0: In this case the resulting equation is automatically satisfied.
C-Integrability Test for Discrete Equations via Multiple Scale Expansions
11
(1)
• Order and θ = 1: If one requires that u1 6= 0, one obtains the dispersion relation
sin (ωσ) = −
sin (κh) σ
.
h
(A.1)
• Order 2 and θ = 0: We obtain the evolution
σ
(0)
(0)
∂m1 u1 − ∂n1 u1 = 0,
h
(A.2)
.
(0)
which implies that u1 depends on the variable ρ = hn1 + σm1 .
• Order 2 and θ = 1: Taking into account the dispersion relation (A.1), we have
σ
cos (κh)
(1)
(1)
(0) (1)
2 κh
u1 u1 = 0,
∂m1 u1 −
∂n1 u1 − 2 sin
cos (ωσ)
h
2
(A.3)
(1)
which implies that u1 has the form
Z ρ
0
(1)
(0)
0
u1 = g (ξ, mj , j ≥ 2) exp δ
u1 ρ dρ ,
ρ0
.
where ξ = hn1 +
cos(κh)
cos(ωσ) σm1 ,
.
δ=
2 sin2 (κh/2)
, (A.4)
[cos (κh) − cos (ωσ)]
g is an arbitrary function of its arguments going to zero as
(0)
ξ → ±∞ and ρ0 , by a proper redefinition of g, can always be chosen to be a zero of u1
(0)
as when ρ → ±∞, u1 → 0 so that there will exists at least one zero.
• Order 2 and θ = 2: Taking into account the dispersion relation (A.1), we have
(1) 2
e−iκh − 1 h
(2)
u2 =
u
.
2 [cos (κh) − cos (ωσ)] 1
• Order 3 and θ = 0: We have
σ
(0)
(0)
(0)
(0) (1) 2
∂m1 u2 − ∂n1 u2 = −∂m2 u1 − 2σ sin2 (κh/2) ∂n1 + 2hu1 u1 .
h
(A.5)
(A.6)
(0)
By equation (A.2), the term ∂m2 u1 is a solution of the left hand side of equation (A.6),
hence it is a secular term. As a consequence we have to require that
σ
(0)
(0)
(0) (1) 2
∂m1 u2 − ∂n1 u2 = −2σ sin2 (κh/2) ∂n1 + 2hu1 u1 ,
h
(0)
∂m2 u1 = 0.
Solving equation (A) taking into account equation (A.4), we obtain
Z ξ
(0)
(1) 2
(0)
(1) 2 0
u2 = f (ρ, mj , j ≥ 2) − hδ cos (ωσ) |u1 | + 2 (1 + δ) u1
|u1 | dξ ,
(A.7)
ξ0
where f is an arbitrary function of its arguments going to zero as ρ → ±∞ and ξ0 is an
arbitrary value of the variable ξ. Let us restrict ourselves for simplicity to the case where
(0)
there is no dependence at all on ρ. If one wants that the harmonic u1 depends on ξ and
(0)
(0)
not on ρ, from equation (A.2) one has that ∂ξ u1 = 0, so that u1 depends on the slow
variables mj , j ≥ 2 only. Similarly we have that ∂ξ f = 0. In this case, in order to satisfy
(0)
(0)
the asymptotic conditions lim uγ = 0, γ = 1, 2, one has to take u1 = f = 0 (unless
ξ→±∞
.
.
we take the fully continuous limit h → 0, hn1 = x1 , σ → 0, σm1 = t1 in which ρ → ξ).
Equation (A.7) then becomes
(1) 2
(0)
u2 = −hδ cos (ωσ) u1 .
(A.8)
12
C. Scimiterna and D. Levi
• Order 3 and θ = 1: Taking into account the dispersion relation (A.1) and the equa(0)
tions u1 = 0, (A.3), (A.5), (A.8), we have
(1) 2 (1)
σ cos (κh)
(1)
(1)
(1)
∂n1 u2 = −∂m2 u1 − iB2 ∂ξ2 u1 − iρ2 u1 u1 ,
h cos (ωσ)
2
2
. hσ h − σ sin (κh)
B2 =
,
2 σ 2 sin2 (κh) − h2 cos (ωσ)
(1)
∂m1 u2 −
. 2hσ sin2 (ωσ/2) [2 cos (κh/2) − cos (3κh/2)] sin (κh/2)
ρ2 = −
[cos (κh) − cos (ωσ)] cos (ωσ)
2
2ihσ sin (ωσ/2) sin (κh/2) sin (κh/2)
−
.
[cos (κh) − cos (ωσ)] cos (ωσ)
(A.9a)
(A.9b)
(A.9c)
(0)
As a consequence of equation (A.3) with u1 = 0, the right hand side of equation (A.9a)
is secular. Hence we have to require that
(1)
σ cos (κh)
(1)
∂n u = 0,
h cos (ωσ) 1 2
(1) 2 (1)
(1)
= B 2 ∂ 2 u + ρ2 u u .
∂m1 u2 −
(1)
i∂m2 u1
ξ 1
1
(A.10a)
(A.10b)
1
(1)
Equation (A.10a) implies that u2 also depends on ξ while equation (A.10b) is a nonintegrable
nonlinear Schrödinger equation, as from the definition (A.9c) we can see that ρ2 is a complex
coefficient. So we can conclude that equation (2.6) is not A1 -integrable.
Let us perform the multiple scale reduction of the Burgers equation (2.1). Equation (2.1) can
.
be always obtained as a semicontinuous limit of equation (2.6) defining the slow times tj = σmj ,
j ≥ 1. In such a way we can use in the present calculation the results presented up above.
• Order and θ = 0: In this case the resulting equation is automatically satisfied.
• Order and θ = 1: Taking the semi continuous limit of equation (A.1), one obtains the
dispersion relation
ω=−
sin (κh)
.
h
(A.11)
• Order 2 and θ = 0: Taking the semi continuous limit of equation (A.2), one obtains
(0)
∂t1 u1 −
1
(0)
∂n u = 0,
h 1 1
(A.12)
.
(0)
which implies that u1 depends on the variable ρ = hn1 + t1 .
• Order 2 and α = 1: Taking the semi continuous limit of equation (A.3), one obtains
(1)
∂t1 u1 −
cos (κh)
(1)
(0) (1)
∂n1 u1 + 2 sin2 (κh/2) u1 u1 = 0,
h
(A.13)
(1)
which implies that u1 has the form
Z
(1)
(1)
u1 = g (ξ, tj , j ≥ 2) exp −
ρ
ρ0
(0)
u1
0
ρ dρ
0
,
(A.14)
.
where ξ = hn1 + cos (κh) t1 , g (1) is an arbitrary function of its arguments going to zero as
(0)
ξ → ±∞ and ρ0 , by a proper redefinition of g, can always be chosen to be a zero of u1 .
C-Integrability Test for Discrete Equations via Multiple Scale Expansions
13
• Order 2 and θ = 2: Taking the semi continuous limit of equation (A.5), one obtains
(2)
u2 =
h
(1)2
u .
1 − eiκh 1
(A.15)
• Order 3 and θ = 0: Taking the semi continuous limit of equation (A.6), one obtains
(0)
∂t1 u2 −
1
(0) (1) 2
(0)
(0)
∂n1 u2 = −∂t2 u1 − 2 sin2 (κh/2) ∂n1 + 2hu1 u1 .
h
(A.16)
(0)
By equation (A.12), the term ∂t2 u1 is a solution of the left hand side of equation (A.16),
hence it is a secular term. As a consequence we have to require that
(0)
1
(0) (1) 2
(0)
∂n1 u2 = −2 sin2 (κh/2) ∂n1 + 2hu1 u1 ,
h
= 0.
∂t1 u2 −
(0)
∂t2 u1
Solving equation (A.17) taking into account equation (A.14), we obtain
(1) 2
(0)
u = f (ρ, tj , j ≥ 2) + hu ,
2
1
(A.17)
(A.18)
where f is an arbitrary function of its arguments going to zero ρ → ±∞.
• Order 3 and θ = 1: For simplicity, from now on we require no dependence on ρ1 so that,
in order to satisfy the asymptotic conditions, it necessarily follows that
(0)
u1 = f = 0.
(A.19)
Taking the semi continuous limit of equations (A.9a), (A.9b), one obtains
(1)
∂t1 u2 −
cos (κh)
(1)
(1)
(1)
∂n1 u2 = −∂t2 u1 − iB2 ∂ξ2 u1 ,
h
. h sin (κh)
B2 = −
.
2
(A.20)
(0)
As a consequence of equation (A.13) (with u1 = 0), the right hand side of equation (A.20)
is secular. Hence we have to require that
cos (κh)
(1)
∂n1 u2 = 0,
h
(1)
(1)
i∂t2 u1 − B2 ∂ξ2 u1 = 0.
(1)
∂t1 u2 −
(A.21a)
(A.21b)
(1)
Equation (A.21a) implies that u2 depends also on ξ while, contrary to equation (A.10b),
equation (A.21b) now is a linear Schrödinger equation, reflecting the C-integrability of
equation (2.1).
• Order 3 and α = 2: Taking into account the dispersion relation (A.11), the fact that
(0)
u1 = 0 and the equations (A.13), (A.15), we have
2
h
(2)
(1)
(1)
(1)
u3 = h
u −
∂ξ u1 u1 .
(A.22)
2
1 − eiκh 2
4 sin (κh/2)
1
(0)
If ∂ρ u1 6= 0, ∂ρ f 6= 0, we have:
Z
(1) .
(2)
u2 = g (n1 , tj , j ≥ 1) exp −
ρ
(0)
u1
0
ρ dρ
0
,
i∂t2 g (1) = B2 ∂ξ2 g (1) ,
ρ0
Z
Z ρ
i
κh
h ρ (0)2 0
(0)
g (2) /g (1) = p (ξ, tj , j ≥ 2) + h 1 + cot
u1 +
u1 dρ −
f ρ0 dρ0 ,
2
2
2 ρ0
ρ0
with p arbitrary function of its arguments going to zero as ξ → ±∞.
14
C. Scimiterna and D. Levi
• Order 3 and θ = 3: Taking into account the dispersion relation (A.11) and equation (A.15),
we obtain
2
(1) 3
h
(3)
u3 =
u1
.
(A.23)
1 − eiκh
• Order 4 and θ = 0: Taking into account equations (A.15), (A.18), (A.19), (A.21b), we
get
1
h (1) 2
(0)
(0)
(1) (−1)
(−1) (1) 2
∂t1 u3 − ∂n1 u3 = h∂ξ
∂ξ u1
− 2 sin (κh/2) u2 u1 + u2 u1
.
h
2
(A.24)
Solving equation (A.24), we obtain
(0)
(1) (−1)
u3 = τ (ρ, tj , j ≥ 2) + h u2 u1
(−1) (1) u1
+ u2
−
(1) 2
h2
u ,
∂
ξ
1
4 sin2 (κh/2)
(A.25)
where τ is an arbitrary function of its arguments going to zero as ρ → ±∞. As usually,
if we don’t want any dependence at all from ρ but only on ξ, in order to satisfy the
(0)
asymptotic conditions lim u3 = 0, we have to take
ξ→±∞
τ =0
(A.26)
(unless we take the fully continuous limit).
• Order 4 and θ = 1: Taking into account the dispersion relation (A.11) and equations
(A.15), (A.18), (A.19), (A.22), (A.25), (A.26), we get
cos (κh)
(1)
(1)
(1)
(1)
(1)
∂n1 u3 = −∂t3 u1 − ∂t2 u2 − B3 ∂ξ3 u1 − iB2 ∂ξ2 u2 ,
h
. h2 cos (κh)
B3 = −
.
6
(1)
∂t1 u3 −
(A.27)
(0)
As a consequence of the equations (A.13), (A.21a) and as u1 = 0, the right hand side of
equation (A.27) is secular, so that
cos (κh)
(1)
∂n1 u3 = 0,
h
(1)
(1)
(1)
(1)
∂t2 u2 + iB2 ∂ξ2 u2 = −∂t3 u1 − B3 ∂ξ3 u1 .
(1)
∂t1 u3 −
(A.28a)
(A.28b)
(1)
The first relation implies that u3 also depends on ξ while the second one, as a consequence
of equation (A.21b), implies that the right hand side is secular, so that
(1)
(1)
i∂t2 u2 − B2 ∂ξ2 u2 = 0,
(1)
(1)
∂t3 u1 + B3 ∂ξ3 u1 = 0.
(A.29a)
(A.29b)
Equation (A.29a), as one can see from the definition (2.16), has a forcing term f2 (2) with
coefficients a = b = 0.
• Order 4 and θ = 2: Taking into account the dispersion relation (A.11) and the equations (A.13), (A.15), (A.18), (A.19), (A.21), (A.22), (A.23), we get
"
#
1
i
cos
(κh/2)
(1)
(1)
(2)
(1)
(1)
2
∂ξ2 u1 u1
u4 = −h3
2 u1 |u1 | +
3
iκh
8 sin (κh/2)
(1 − e )
−
h2
h
(1) (1) (1)2
(1) (1) ∂ξ u1 u2 +
u2 + 2u1 u3 .
iκh
1−e
4 sin (κh/2)
2
(A.30)
C-Integrability Test for Discrete Equations via Multiple Scale Expansions
15
• Order 4 and θ = 3: Taking into account the dispersion relation (A.11) and the equations (A.13), (A.15), (A.19), (A.22), (A.23), we get
2 h
2heiκh
(3)
(1)
(1)
(1)
u4 =
u
3u2 +
∂ξ u1
.
1 − eiκh 1
1 − eiκh
• Order 4 and θ = 4: Taking into account the dispersion relation (A.11) and the equations (A.15), (A.23), we get
3
(1) 4
h
(4)
u4 =
u1
.
iκh
1−e
• Order 5 and θ = 0: Taking into account equations (A.15), (A.18), (A.19), (A.21b), (A.22),
(A.26), (A.25), (A.28), we get
n
1
(0)
(0)
(1) (−1)
(−1) (1)
(1) ∂t1 u4 − ∂n1 u4 = h∂ξ − 2 sin2 (κh/2) u1 u3 + u1 u3 + |u2 |2
h
h2 (1) 4 h
u + ∂ξ u(1) u(−1) + u(−1) u(1)
+ 4 sin2 (κh/2) − 1
1
1
2
1
2
2
2
o
ih2
(−1)
(1)
(1)
(−1) cot (κh/2) ∂ξ u1 ∂ξ u1 − u1 ∂ξ u1
.
(A.31)
+
4
Solving equation (A.31), we obtain
(0)
(1) (−1)
(−1) (1)
(1) u4 = Θ (ρ, tj , j ≥ 2) + h u1 u3 + u1 u3 + |u2 |2
n
(1) 4 o
h2
(1) (−1)
(−1) (1) 2
∂
u
u
+
u
u
+
4
sin
(κh/2)
−
1
h u 1 +
ξ
1
2
1
2
4 sin2 (κh/2)
ih3 cos (κh/2)
(−1)
(1)
(1)
(−1) −
∂ξ u1 ∂ξ u1 − u1 ∂ξ u1
,
(A.32)
3
8 sin (κh/2)
where Θ is an arbitrary function of its arguments going to zero as ρ → ±∞. As usually,
if we don’t want any dependence from ρ but only on ξ, in order to satisfy the asymptotic
(0)
conditions lim u4 = 0, we have to take
ξ→±∞
Θ=0
(unless we take the fully continuous limit).
• Order 5 and θ = 1: Taking into account the dispersion relation (A.11) and equations
(0)
u1 = f = τ = Θ = 0, together with the equations (A.15), (A.18), (A.22), (A.23), (A.25),
(A.30), (A.32), we get
cos (κh)
(1)
(1)
(1)
(1)
(1)
(1)
∂n1 u4 = −∂t4 u1 − ∂t3 u2 − ∂t2 u3 − iB2 ∂ξ2 u3 − B3 ∂ξ3 u2
h
h
i
(1)2 2 (−1)
(1) (1) 2
(−1)
(1) 2
4 (1)
+ u ∂n u
,
(A.33)
+ iB4 ∂ u + ζ u ∂n u
+u
∂n u
(1)
∂t1 u4 −
ξ 1
. h3 sin (κh)
B4 =
,
24
1
1
1
1
1
1
1
1
1
. h [1 + cos (κh) + 3i sin (κh)]
ζ=
.
eiκh − 1
(0)
As a consequence of the equations (A.13), (A.21a), (A.28a) and of u1 = 0, the right hand
side of equation (A.33) is secular, so that
(1)
∂t1 u4 −
cos (κh)
(1)
∂n1 u4 = 0,
h
16
C. Scimiterna and D. Levi
(1)
(1)
(1)
(1)
(1)
∂t2 u3 + iB2 ∂ξ2 u3 = −∂t4 u1 − ∂t3 u2 − B3 ∂ξ3 u2
h
i
(1) 2
(1) 2
(1)2
(−1)
(1)
(1) (−1)
+ u1 ∂n21 u1
+ iB4 ∂ξ4 u1 + ζ u1 ∂n1 u1 + u1
∂n1 u1
.
(A.34)
(1)
The first relation tells us that u4 depends on ξ too while in the second one, as a consequence of equations (A.21b), (A.29a), the first our terms in the right hand side of
equation (A.34) are secular, so that
h
i
(1) 2
(1) 2
(1)2
(−1)
(1)
(1)
(1) (−1)
+ u1 ∂n21 u1
∂n1 u1
, (A.35a)
∂t2 u3 + iB2 ∂ξ2 u3 = ζ u1 ∂n1 u1 + u1
(1)
(1)
(1)
(1)
∂t3 u2 + B3 ∂ξ3 u2 = −∂t4 u1 + iB4 ∂ξ4 u1 .
(A.35b)
As a consequence of equation (A.29b), the right hand side of equation (A.35b) is secular
so that
(1)
(1)
∂t3 u2 + B3 ∂ξ3 u2 = 0,
(1)
(1)
∂t4 u1 − iB4 ∂ξ4 u1 = 0.
Equation (A.35a), as one can see from the definition (2.17) and taking into account
that a = b = 0, has a forcing term f2 (3) that respects all the A3 C-integrability conditions (2.18).
Acknowledgements
The authors have been partly supported by the Italian Ministry of Education and Research,
PRIN “Nonlinear waves: integrable finite dimensional reductions and discretizations” from 2007
to 2009 and PRIN “Continuous and discrete nonlinear integrable evolutions: from water waves
to symplectic maps” from 2010.
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